drift shell bifurcation near the dayside magnetopause in realistic magnetospheric magnetic fields

8/3/2019 Drift shell bifurcation near the dayside magnetopause in realistic magnetospheric magnetic fields

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Drift shell bifurcation near the dayside magnetopause in realistic

magnetospheric magnetic fields

Yifei Wan,1 Stanislav Sazykin,1 Richard A. Wolf,1 and M. Kaan ztrk2

Received 24 February 2010; revised 23 April 2010; accepted 13 May 2010; published 2 October 2010.

[1] We study trapped energetic particles in the terrestrial magnetosphere undergoingdrift shell bifurcation in the magnetic field lacking northsouth and eastwest symmetry.Drift shell bifurcation occurs near the dayside magnetopause, where, due to thesolar wind compression, the field strength has a local maximum near the equatorial

plane. As a result, a charged particle may become temporarily trapped in one of thehemispheres while traversing the region. Although this phenomenon has been knownfor a long time, only recently were the associated second invariant changes quantifiedfor the magnetic field with northsouth and eastwest symmetry. Here we show thatif the magnetic field lacks such symmetry, the effect is more significant. We calculatechanges to the second invariant of keV to MeV electrons in Tsyganenko magneticfields with nonzero interplanetary magnetic field (IMF) BY component. The changes areon the order of the invariant itself, and thus, this effect is much larger than for the caseof symmetric magnetic field (when the particle gyroradius is much less than themagnetospheric scale length). We also quantify the effect for different values of thesolar wind dynamic pressure, IMF BZ component, and the Dst index with theTsyganenko magnetic field T02. We find that Dst has no noticeable role, while largersolar wind ram pressure increases the second invariant changes. We verify our calculations

by numerical integration of the guiding center drift equations and discuss properties ofdifferent versions of these equations.

Citation: Wan, Y., S. Sazykin, R. A. Wolf, and M. K. ztrk (2010), Drift shell bifurcation near the dayside magnetopause inrealistic magnetospheric magnetic fields, J. Geophys. Res., 115, A10205, doi:10.1029/2010JA015395.

1. Introduction

[2] In the typical magnetospheric magnetic field config-uration, trapped particles of sufficiently high energies willexecute gradient and curvature drifts on closed orbits whilegyrating and bouncing between the hemispheres in theconverging magnetic field (mirror) geometry. Separation ofcharacteristic time scales of the three quasiperiodic motionsleads to the most common mathematical description (useof adiabatic invariants) to study the dynamics of such par-ticles [e.g., Northrop, 1963; Walt, 1994]. Processes thatthen cause the changes of the adiabatic invariants must

be accounted for separately. In this paper, we analyzeone particular not very wellknown mechanism leading to

changes of the second and third adiabatic invariants for particles drifting through the narrow region around thedayside magnetopause, namely, drift shell bifurcation (DSBfor brevity).

[3] Trajectories of particles drifting in a static magneticfield without electric field could be obtained by numericalintegration of the full relativistic NewtonLorentz equation,

d

dtmv qv B: 1

[4] Here g = (1 v2 / c2)1/2 is the relativistic factor, mand q are particles mass and charge, v is the velocity vector,and B is the local magnetic field. However, for reasons to beexplained in section 4, we adopt the socalled guiding centerapproach in this work. The phenomenon of drift shell

bifurcation is illustrated later in the left column of Figure 4,

which shows the guiding center trajectories of an electron inthe Earths magnetic field described by the Tsyganenko T02model (also commonly referred to as T01_01) [Tsyganenko,2002a, 2002b]. The electron drifts around the Earth whileexecuting the familiar bouncing motion about the equatorial

plane (except on the dayside when it is trapped into one ofthe hemispheres), which typically can be taken as a surface ofminimum magnetic field strength. The last fact is often stated

by saying that the curve of magnetic field strength versus thedistance along the magnetic field line has a U shape during thehalfbounce period. The surface mapped out by the trajectoryis called a drift shell.

1Physics and Astronomy Department, Rice University, Houston, Texas,USA.

2Department of Information Systems and Technologies, YeditepeUniversity, Istanbul, Turkey.

Copyright 2010 by the American Geophysical Union.01480227/10/2010JA015395

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, A10205, doi:10.1029/2010JA015395, 2010

A10205 1 of 17
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[5] Let us denote the first and second adiabatic invariantsas m and J. If the energy is conserved as will be in our case,the mirror field Bm can be used in place ofm,

Bm

mv22

2;

2

and the second invariantJcan be replaced by the geometricsecond invariant I,

I Bm J2p

Zsm2

sm1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 B s

Bm

sds; 3

where p is the momentum, B(s) is the magnetic field strengthalong a field line, and sm1 and sm2 are the locations of themirror points. In a limited region before noon, an electron (ifthe mirror field Bm is sufficiently small) undergoes a rapid(over a couple of bounce periods) transition (bifurcation) and

becomes trapped in one of the hemispheres. There is anotherpoint past noon where the electron returns to its bouncemotion about the equatorial plane. As can be seen from theamplitude of the bounce motion, the second invariant I haschanged and the electron is now on a different drift shell.

[6] Trajectories of the type plotted in Figure 4 are oftencalled Shabansky orbits [Shabansky, 1971]. This phe-nomenon has been known for a long time [e.g., Northrop andTeller, 1960; Mead, 1964; Shabansky and Antonova, 1968;

Roederer, 1970; Delcourt and Sauvaud, 1999; ztrk andWolf, 2007] and is a result of the magnetic field havinga region on the dayside where, due to compression of the

magnetosphere by the dynamic pressure of the solar wind,magnetic field strength profiles along field lines have aW shape, as illustrated in Figure 1 for a typical T02 magneticfield configuration. In other words, there are two minima thatare off the equatorial plane, and it is possible for particles to

become trapped and bounce about one of them.[7] It can be argued [e.g., Shabansky, 1971; ztrk and

Wolf, 2007, and references therein] that the first adiabaticinvariant is conserved along the trajectory, but we later verifyin section 7.2 through fullparticle tracing that the first adi-abatic invariants of the electrons we study are indeed con-served. Of particular interest is the change in the secondinvariant after a particle drifts through the region near thedayside magnetopause with a mirror field weaker than thelocal nearequatorial maximum (see Figure 1). Calculation ofsuch nonadiabatic change is the subject of this paper. Previ-ously, ztrk and Wolf [2007] extensively analyzed driftshell bifurcations and the changes of the second invariantIfor a magnetic field configuration that was a superposition of

two parallel dipoles. Their results were stated for magneticfields with northsouth and eastwest symmetry. In thesymmetric case, the change DI in the second invariant re-sulting from one passage across the day side scales with thegyroradius r and is typically a few rs (small comparedto initial I). When the result is expressed as a radial diffusionin L with radiation belt particles in mind, the value of thediffusion coefficient becomes comparable to that due tomagnetic field fluctuations for radiation belttype electrons ofenergies only in excess of 10 MeV. In other words, the effectDSB has on violation of the second invariant conservation inthe symmetric case is likely small.

Figure 1. Magnetic field strength along the field plotted for three magnetic local times for the T02model with parameters solar wind ram pressure PDYN = 2 nPa, zero tilt angle, IMF BY = 10 nT, IMF

BZ = 10 nT, Dst = 0. All three field lines exhibit typical W shapes. Curves labeled dawn and duskhave their equatorial crossings at 10 MLT and 14 MLT, respectively. All three field lines cross the equa-torial plane at the distance 10.1 RE.

WAN ET AL.: DRIFT SHELL BIFURCATION IMF BY A10205A10205

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[8] At the same time, it was pointed out (but not analyzed)by ztrk and Wolf [2007] that if the magnetic field has adawndusk asymmetry, one can expect a change in secondinvariant simply due to the asymmetry in the magnetic field,and that the effect may be of zeroth order in gyroradius andindependent of energy and bounce phase. It is the purpose of

this paper to extend the analysis to this case and characterizeand quantify the second invariant changes of particles under-going DSB in realistic magnetic fields. One obvious reasonfor such an asymmetry is penetration of the y component(in the GSM coordinates) of the interplanetary magnetic fieldinto the magnetosphere.

2. Assumptions

[9] In this work, we assume that the external magneticfield is given by the semiempirical magnetic field modelT02 [Tsyganenko, 2002a, 2002b] parameterized by the IMF

BY and BZ values, solar wind dynamic pressure PDYN, Dstindex, and the value of the dipole tilt angle. We further

assume that the tilt angle is zero, and that the intrinsicmagnetic field of Earth is a dipole with no tilt. Unless statedotherwise, we use a magnetic field with IMF BZ = 5 nT,

PDYN = 2 nPa, and Dst = 0. Constancy of magnetic field intime allows us to separate the effects of nonzero BY from theradial diffusion due to field fluctuations. Electric field is alsoassumed to be zero since it is not essential for this analysis.One prerequisite of the study is that the first invariantm isconserved. This places a restriction on the gyroradius, thusexcluding particles with very high energies. We treat elec-trons only; for protons, the results will be the same. However,the following facts should be considered: (1) they gradient

and curvature drift around the Earth in the opposite direc-tion; (2) they have larger gyroradii, which leads to a stricterrestriction on the conservation of the first adiabatic invariant;thus, protons subject to the investigation of the DSB effect aresupposed to have lower energy ranges than electrons.

3. The nature of the Second Invariant Changesin the Case ofBY 0

[10] Magnetic field strength curves such as the dashed anddashdotted ones in Figure 1 become asymmetric withrespect to their equatorial crossing points if BY 0. Forinstance, if IMF BY < 0 (directed dusk to dawn), then on thedawn side, the local minimum field strength in the NorthernHemisphere BminN is smaller than its counterpart BminS inthe Southern Hemisphere; on the dusk side, BminN > BminS.The relations between the two local minima are just theopposite for positive (dawntodusk) IMF BY conditions.

[11] Figure 2 shows a sample calculation of profiles of the

magnetic field strength along an electron trajectory acrossthe bifurcation region. Shown are 5 points in time as theelectron drifts from dawn across noon onto the afternoonlocal time sector under negative IMF BY condition. Whenthe local maximum value (near the equatorial plane) of agiven magnetic field line exceeds the mirror field strength

Bm, which is conserved along the path, let us define twoquantities

In Bm Zsn2

sn1


Bm

sds 4

Figure 2. A sequence ofsnapshots of profiles of magnetic field strength along field lines at 5 pointsalong the trajectory (time goes left to right) of an electron drifting through the bifurcation region. Parti-cles parameters are Bm = 55 nT, initial second invariantIi = 3 RE. IMF BY = 5 nT for this case.


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and

Is Bm Zss2

ss1


Bm

sds; 5

representing the second adiabatic invariants in the Northernand Southern hemispheres, respectively. Here sn1 and sn2correspond to the mirror points in the Northern Hemispherewhile ss1 and ss2 are in the Southern Hemisphere. A smaller

B(s) contributes more to the integral for a given mirror fieldBm, which should of course be larger than the local mini-mum and smaller than the equatorial maximum to make theintegral meaningful. So on the dawn side, In > Is for thesame field line, while on the dusk side, In < Is. These rela-tions could also be illustrated by the contour plots of In and

Is projected to the equatorial plane (Figure 3). For negativeIMF BY, In has larger values on the dawn side while Is islarger on the dusk side.

[12] In our study trapped energetic electrons drifting

around the Earth keep their second adiabatic invariantsconserved except when they come to a bifurcation field linenear the dayside magnetopause, where the equatorial maxi-mum just exceeds the mirror field strength, and get trappedinto one of the hemispheres. It will cause the sudden decreaseof the second adiabatic invariant. Conserving In or Is anddrifting past the local noon, the electrons will face the other

bifurcation field line and escape the confinement with asudden increase of the second adiabatic invariant. But dueto the existence of nonzero IMF BY, the magnitudes of thesudden changes will be different, which explains the netchange of the total second adiabatic invariant after drifting

across the bifurcation region. The change turns out to be ofthe same order of the initial second adiabatic invariant, as wewill show below.

4. Guiding Center Tracing Results

[13] Since solving (1) numerically is computationally veryexpensive and is susceptible to numerical errors, a simplerapproach to analyzing effects of DSB on the second invariantchanges in the case of nonzero BY is to trace particle trajec-tories numerically by solving a version of the guiding centerequations [e.g., Northrop, 1963; Walt, 1994; Brizard andChan, 1999; Tao et al., 2007]. Of course the conservationof thefirst adiabatic invariant should first be ensured, which isthe case in our study. For reasons explained later in section 7,we chose to solve the form of equations derived [Brizard andChan, 1999] from the Hamiltonian formalism,

dR

dt v

k

B*

B*k

qB*k

b

rB;

6

dvkdt

2mB*k

B* rB; 7

where B* = B + gmv||r b /q,B||* = B* b =B + gmv||b (r b) / q, b is the unit vector of the magnetic field, v|| is thevelocity component parallel to B, and Ris the position of the

particles gyrocenter.[14] In Figure 4, we show four qualitatively different

types of solutions that can be obtained by numerically

Figure 3. Contour plots ofIn and Is (equations (4) and (5)) in the dayside equatorial plane under15 nTIMF BY. The mirror field strength is 55 nT.


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Figure 4. Four cases of guiding center trajectories in the magnetic field with IMF BY = 5 nT illustratingfour types of drift shell bifurcation (the differences are initial conditions as listed in Table 1). Electrons arelaunched from the equatorial plane at 6 MLT with initial velocity Vand initial pitch angle a. II and IF areinitial and final second invariants, respectively. (left) Trajectories. (right) Second invariants along trajectories.


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integrating equations (6)(7) for the same magnetic fieldusing the embedded CashKarp RungeKutta integrationscheme [Press et al., 1992]. We mostly choose the relativeaccuracy of 106 in the adaptive stepper. The associatedvariations of the second invariant I differ due to differentinitial conditions, which are listed in Table 1.

[15] The two obvious cases for the second invariantchanges are shown in (b) and (c), where the electron un-

dergoes bifurcation twice, on both sides of the local noon,and becomes trapped in the southern (b) or northern (c)hemisphere. During the electrons drift motion, its secondadiabatic invariant is conserved except near the bifurcationfield lines. Consider Figure 2 (obtained from a case similarto the one of (b)). The electron with initial second adiabaticinvariant In1 + Is1 gets trapped into the Southern Hemi-sphere at the dawn side bifurcation line. It then drifts acrossthe bifurcation region, keeping its invariant Is1, until itcomes to the dusk side bifurcation line and returns to the

Northern Hemisphere with second adiabatic invariantIn2 +

Is2. Since Is1 = Is2 and In1 > In2, the total second adiabaticinvariant will be changed by the amount of

DI In2 In1: 8

Likewise, an electron trapped in the Northern Hemispherewill experience a change in the second invariant by theamount of

DI Is2 Is1: 9

There are two more, less obvious cases, when the change ofthe second invariant results from bifurcating only once ((a)and (d)). These are two special cases when the electronsinitial second adiabatic invariant is small. The electron willexperience only one bifurcation point in the bifurcationregion and get its second adiabatic invariant changed onlyonce. For an electron with small initial second adiabaticinvariant (Figure 5), when it drifts to the location where themagnetic field profile first becomes W shape, the localmaximum field magnitude is already larger than the mirrorfield value (so there is no bifurcation point on the dawnside) and the electron is trapped around the field minimum

in the Northern Hemisphere. It will thus only experience one bifurcation point on the dusk side, increasing its secondadiabatic invariant by the amount of

DI Is2: 10

The other special case (Figure 6) happens when the electronbifurcates on the dawn side with a small second invariantIs1. If there is no corresponding bifurcation point (becausewhen Is2 = Is1 the local maximum field strength is largerthan the mirror field magnitude) on the dusk side, the electron

Table 1. Four Cases for Which Guiding Center Calculations arePresented in Figure 4a

Case YGSM(RE) V(c) a () II (RE) IF (RE) DI (RE) Bm (nT) E (keV)

(a) 7.5 0.6 75.4 0.23 1.04 0.81 60.0 127.7(b) 8.5 0.5 4 9.0 2.10 1.68 0.42 65.0 79.1(c) 8.0 0.3 60.0 1.03 2.07 1.04 60.5 24.7(d) 8.0 0.5 6 0.5 0.99 0.20 0.79 60.0 79.1

a

Electrons are launched from the equatorial plane at 6 MLT with initialradius |YGSM|, initial velocity Vand initial pitch angle a. II and IF are initialand final second invariants, respectively. DI is the change of the secondinvariant after drift shell bifurcation. Bm and E are the mirror fieldmagnitude and energy, respectively.

Figure 5. Snapshots of profiles of magnetic field strength along field lines for the case with a bifurcationpoint only on the dusk side. Bm = 55 nT, initial second invariant Ii = 0.3 RE. IMF BY = 5 nT.


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will drift out of the bifurcation region with its second adia-batic invariant changed by the amount of

DI In1: 11

5. Phase

Space Diagrams[16] Another useful way to look at the dynamics of the

particle undergoing drift shell bifurcation and nonadiabaticchange of its second invariant is to invoke phasespace dia-grams, orpq diagrams, widely used in nonlinear mechanics.In Figure 7, we presentpq diagrams for the case (c) (Figure 4and Table 1). Closed curves represent quasiperiodic motion.The area enclosed by each curve is proportional to the secondinvariant I. It is easy to see how the character of motionchanges at the two bifurcation points and that the final areaenclosed in the curve after the particle drifts around the Earthand comes back to the starting point has changed.

6. Analysis Based on Magnetic Field Geometry[17] Numerically solving guiding center equations allows

for detailed analysis of particle dynamics. However, as waspointed out by ztrk and Wolf [2007], under some addi-tional assumptions listed in section 2, changes in the secondinvariantDIcan be estimated for a particle of given mirrorfield strength Bm and initial second invariant I by consid-ering properties of the magnetic field only. The results thencan be compared with sample calculations obtained fromguiding center tracing to ensure consistency. In this section,we describe the procedure for evaluating the second invariantchanges for any values ofBm and I(defining drift shells) of

particles entering the bifurcation region and present our mainresults. An obvious advantage of this procedure is its ease ofuse, as no trajectory tracing is necessary.

6.1. Procedure to Compute the Second InvariantChanges

[18] Before we continue, we want to clarify the definitions

of three terms that are used throughout the paper. Thebifurcation region is the area near the dayside magnetopausewhere magnetic field strength along the field line has a localmaximum near the equator. This also implies that a localminimum exists on each side of the equator. The bifurcationfield line is one on the particles drift path where the nearequator magnetic field strength Bpeak just equals the mirrorfield value Bm. The bifurcation point is defined as its equa-torial crossing point. To sort out the bifurcation points, weneed to define two sets of contours. In Figure 8, we plot aset ofBpeak contours C(Bpeak) at the equatorial crossings ofthe magnetic field lines for5 nT IMF BY. As particles driftcloser to local noon, Bpeak exceeds Bm, which leads to parti-cles being trapped into one of the hemispheres. In addition,

we need contours C(I(Bpeak)), where Bm in the definition(3) of the total second invariant I is replaced by the localmaximum fieldBpeak. Figure 9 illustrates theI(Bpeak) contoursunder the same IMF conditions. It is noteworthy that theintersection of C(Bpeak) and C(I(Bpeak)) (Figure 10) onlydefines the bifurcation point when the particle enters the

bifurcation region. At this critical point, the particle decideswhether to continue its drift in the northern or SouthernHemisphere. After that, its second invariant is conserved untilit leaves the bifurcation region.I(Bpeak) should be replaced by

In(Bpeak) orIs(Bpeak) when searching for the other bifurcationpoint, depending on the hemisphere it was trapped in. It is

Figure 6. Snapshots of profiles of magnetic field strength along field lines for the case with a bifurcationpoint only on the dawn side. Bm = 55 nT, initial second invariantIi = 1.2 RE. IMF BY = 5 nT.


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Figure 7. Electron phase space trajectories in one bounce period between mirror points at different mag-netic local times along the drift trajectory in Figure 4c. Here p is the (canonical) parallel momentum, andconjugate coordinate q is the distance along the magnetic field line. The area inside the curve is pro-

portional to the second invariantI. Particle traverses from left to right and top to bottom.


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then straightforward to evaluate the second invariant changesbased on the values at the corresponding bifurcation points.

6.2. Results

[19] When we apply the procedure outlined in section 6.1for a number of values of the mirror field and the secondinvariant, we obtain a set of curves shown in Figure 11. TheFigure 11 (top) is for our nominal magnetic field configu-ration with IMF BY = 5 nT. In general, for each I valuethere are two possibilities forDI, corresponding to trapping inthe Northern Hemisphere (solid lines) or Southern Hemi-sphere (dashed lines) depending on the initial conditions.The most important conclusion that one should make fromFigure 11 is that typical changes in Iare of the same order ofmagnitude as the initial I, thus the effect is significant. Tomake sure that this effect is not spurious and specific to this

particular value of IMFBY, we present results in the other twopanels of Figure 11 for the cases of10 and 15 nT IMF BY.

These results indicate that the changes of the second invariantare large for all the nonzero BY cases we study.

6.3. Dependence on IMF BZ, Dst, and PDYN

[20] To estimate how sensitive this effect is to differentTsyganenko T02 model configurations, we set the defaultvalues as solar wind dynamic pressure PDYN =2nPa,Dst= 0,IMF BY = 5 nT, and BZ = 5 nT (Figure 11, top) and changeonly one parameter at a time. The effect of varying values ofIMF BY was already shown in the previous subsection.Generally, stronger negative BY causes the absolute value ofDIto increase, but it isno longer true whenBmbecomes large.

Figure 8. Contours of the nearequator maximum mag-netic field strength Bpeak under5 nT IMF BY.

Figure 9. Contour plot ofI(Bpeak) under the same IMF con-ditions as Figure 8.

Figure 10. Contour plots of Bpeak and I(Bpeak). The inter-sections define the bifurcation points where particles enterthe bifurcation region.


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[21] At a fixed Bm level, the absolute value ofDIis largerwhen PDYN (Figure 13, right) is larger but decreases whensouthward (negative) IMF BZ (Figure 12) is stronger, whileDst (Figure 13, left) only has a small influence on thechange of the second invariant. In these plots, Bm values arechosen only when drift shell bifurcation can happen at thoselevels.

6.4. Consistency With Guiding Center Tracing Results[22] Having at our disposal the program to trace guiding

center trajectories as described in section 4, we compareresults fromthe twoapproaches in Figure 14.Superimposed onthe curves obtained with the procedure outline in section 6.1are four crosses that represent solutions obtained by solvingthe guiding center equations from section 4. It can be seenthat, for the four cases of guiding center calculations, theresults are in excellent agreement with calculations in thissection. To get points from solving (6)(7) that cover a largerange of mirror fields and second invariants requires a trialand error approach in choosing the initial conditions; there-fore, we only list four representative cases, although in ourexperience the agreement is always good.

7. Discussion

[23] Before summarizing the results, there are several pointsthat need to be addressed and are specific to this particularsituation.

7.1. Guiding Center Equations

[24] First, we would like to comment on the choice of theguiding center approximation equationsused in section 4. It isknown [e.g., Walt, 1994; ztrk and Wolf, 2007; Kim et al.,2008] that the standard textbook guiding center approxi-mation equations derived by Northrop [1963],

dRdt

vk BB

qB2

B rB mv2kqB2

B k; 12

dvkdt

2mB

B rB; 13

where m = g2mv?2 / (2B)(m / g is the relativistic magnetic

moment) and k = (b r)b, are not sufficiently accurate whennumerically tracing particles. Specifically, Walt[1994] statesthat the second equation, (13), needs to be more accuratefor numerical orbit tracing, while ztrk and Wolf[2007] and

Kim et al. [2008] choose (6)(7) because these equationsinclude higherorder terms and thus are supposed to be moreaccurate. Brizard and Chan [1999] [also see Tao et al., 2007]

state that equations (6)(7), derived from the Hamiltonianapproach [Littlejohn, 1983; see also review by Cary and

Brizard, 2009, and references therein], can be shown tocontain the original Northrop equations (12)(13) when

Figure 11. Changes in the second invariant for a number oflevels of Bm (its value being labeled by the number at theright end of each curve; in unit nT) and I computed for(top) IMF BY = 5 nT, (middle) IMF BY = 10 nT, and (bot-tom) IMF BY = 15 nT. Solid lines, particle trapped in the

Northern Hemisphere (also labeled by N); dashed lines, inthe Southern Hemisphere (also labeled by S).


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neglecting phasespace volumepreserving terms (such as thedifference between B||

* and B). Here we attempt to clarify thesituation.

[25] When numerically integrating the standard Northropequations with the Tsyganenko 02 model, we experiencedthe nonconservation of the second invariant (even whenthe electron was not at the bifurcation lines), which might

be expected if the equations are not sufficiently accurate.

However, we also found nonconservation of the mirror fieldBm (as shown in Figure 15), which is an indication of thenonconservation of energy when integrating the equations. Itshould be noted that our numerical integration was carriedout with very high accuracy to minimize numerical errors.

Nonconservation ofBm is more troublesome and makesequations (12)(13) unsuitable for this study. By comparison,here we show explicitly the two extra terms contained in thesecond BrizardChan equation and how the problem could

be solved by adding these terms to the standard Northropequations.

[26] If we take (6)(7) and expand B* and B||* in powers

of " = m/q (proportional to the gyroradius), equation (6)becomes

dR

dt vk

B*

B*k

qB*kb rB

vkB mvk

qr b

B mvkq

b r b

q

1

B mvkq

b r b b rB

vkB

B mvkq

r b

1 mvkqB

b r b O "2 !

qB

1 mvjjqB

b r b O "2

!

b rB

vkB

B mv2k

qBr b vk

BBmvk

qBb

r b qB

b rB O "2

vkB

B mv2k

qBr b b b r b

qBb rB O "2 :

It is equivalent to equation (12) after omitting O("2) termsbecause

b k b b r b b r b b r b b b r b :

Figure 12. Changes in the second invariant for a number oflevels of Bm (its value being labeled by the number at theright end of each curve; in unit nT) and I computed for(top) IMF BZ = +5 nT, (middle) IMF BZ = 10 nT, and (bot-tom) IMF BZ = 15 nT. Solid lines, particle trapped in the

Northern Hemisphere (also labeled by N); dashed lines,in the Southern Hemisphere (also labeled by S). IMF

BY = 5 nT for all three cases.


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Equation (7) could be rewritten as

dvkdt

2mB*k

B* rB

2m

1

B

mvk

qb

r b

B mvk

qr b rB

2mB

1 mvkqB

b r b O "2 !

B mvkq

r b

rB

2mB

B rB mvkq

r b rB

mvkq

b r b b rB !

O "2 :

It contains two more terms of the order of " when compared to equation (13). These two extra terms could beneglected when the magnetic field is curl free, such as adipole or doubledipole magnetic field. When r B = r (Bb) = Br b + rB b = 0, r b = (rB b) /B, so bothterms vanish. Numerically, we did not encounter the problemof energy nonconservation when tracing particles underdipole or doubledipole models. However, for Tsyganenkomagnetic fields, the external magnetic field induced by dif-ferent current systems breaks the curlfree condition. Theabsence of these two terms leads to unphysical results.Adding them to the standard Northrop equations, we gotthe same simulation results (Figure 16) as those of theBrizardChan equations, which were already shown pre-viously (Figure 4c).

[27] It could also be shown that the BrizardChan equa-tions are equivalent to the guiding center equations extendedto higher order by Northrop and Rome [1978] (specificallyequations (3) and (5) in that paper) if we neglect the relativ-

Figure 13. Changes in thesecond invariant for a number of levels ofBm (its value being labeled by the num-ber at the right end of eachcurve; in unit nT) andI: (left) for two levels of Dst,25nTand 50 nT; (right) fortwo values ofPDYN of 5 and 8 nPa. Solid lines, particle trapped in the Northern Hemisphere (also labeled byN), dashed lines, in the Southern Hemisphere (also labeled by S). IMF BY = 5 nT for all cases.


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istic effect and replace m with M= M0 + "M1. However, theevaluation of the latter requires the detailed information of a

particles initial position and velocity, while initial guidingcenter position, velocity magnitude, and pitch angle are suf-ficient for solving the BrizardChan equations.

7.2. Validity of Guiding Center Approximation

[28] The other aspect of numerically integrating guiding

center equations under the conditions of nonzero BY thatshould be pointed out is theallowed values of initial velocities(or total energy). The results presented in this paper do not

depend on the total energy; thus one might be tempted tochoosehigher values of initial velocity to speed up integrationof (6)(7), which, with the appropriate accuracy, are com-

putationally slow. It could be argued that as long as thegyroradius of the particle is small with respect to the fieldvariation length scale, the guiding center approximation isapplicable. More specifically, the length scale of magneticfield change experienced by a charged particle during the

gyration period should be much larger than its gyrationradius, which also implies the conservation of the first adia-batic invariant. The often used criterion [Sergeev et al., 1983;

Figure 14. Comparison of results obtained with geometry calculations (lines) and four cases of guidingcenter drift trajectory solutions (four crosses). The curves are the same as in Figure 11 (top), while thecrosses are from the four cases in Table 1 and Figure 4.

Figure 15. Results of tracing an electron with initial position (0, 8, 0) RE in GSM coordinates, velocity0.3c, pitch angle 60 under Tsyganenko 02 model using the standard Northrop equations (12)(13). (left)The second invariant as a function of time along the electrons trajectory. (right) The magnetic fieldstrength the electron experiences along its trajectory, with the top envelope corresponding to the mirrorfield magnitude.


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Bchner and Zelenyi, 1989; Delcourt et al., 1996; Wolf et al.,2009] is

2 Rc;minac;max

) 1

whereRc,min is the minimum curvature radius of the magneticfield line and ac,max is the maximum gyration radius along thefield line. The critical values of for the conservation of thefirst adiabatic invariant suggested by Sergeev et al. [1983]and Delcourt et al. [1996] are

ffiffiffi8

pand 3, respectively. There

are also other ways to measure the nonadiabatic scattering ofthe first adiabatic invariant, such as discussed for ions by

Anderson et al. [1997].[29] In our study of the drift shell bifurcation effect, the

existence of nonzero interplanetary magnetic fields decreases

the curvature radius of magnetic field lines. We could not usetoo energetic particles, whose gyration radii become comparable to the curvature radii, especially near the plasmasheet on the night side, to carry out guiding center tracing.This was somewhat unexpected as this type of scattering ofthe first invariant is usually associated with highly stretchedmagnetotailplasma sheet field lines and not the inner mag-netosphere (L < 8). Yet too energetic particles should not betraced through guiding center equations, since the assumptionfor guiding center equations is violated.

[30] We tested full particle tracing for different particlevelocities by integrating the relativistic NewtonLorentzequation (1) under the same solar wind and IMF condi-tions as in previous guiding center numerical simulations.

Figure 17 (top) shows the magnetic field magnitude along the0.99c electrons trajectory and the corresponding values.Instead of showing the value for each magnetic field line,here we calculate the instant value (square root of the ratioof the curvature radius to the gyration radius) for each traced

point along the electrons trajectory. At the bottom ofFigure 17, we show solutions of (1) but now for an electronwith 0.9c velocity magnitude, all other conditions being thesame. Comparison of the two cases suggests a critical value of 9 to ensure good conservation of the first adiabaticinvariant, as implied by the conservation of the mirror fieldmagnitude (refer to the definition of the first adiabatic invari-

ant). We use this value as our guideline for selecting electronsand verifying the validity of guiding center simulation.

[31] We are especially careful on this topic in our studybecause of the existence of nonzero interplanetary magneticfields. For magnetic fields with eastwest and northsouthsymmetries, much more energetic particles could be used inguiding center tracing.

7.3. Probability of Being Trapped in a GivenHemisphere

[32] It would be of interest to estimate the average chan-ges of the second invariants during drift shell bifurcationunder nonzero IMF BY, which requires knowledge of the

probability of the particle becoming trapped in a givenhemisphere for averaging purposes. The change of thesecond invariant (for given initial Iand Bm) depends on the

hemisphere the particle is trapped in, which is determined bythe bounce phase of the particle on the bifurcation field line.Shabansky [1971] hypothesized that the probability of the

particle entering the northern or Southern Hemisphere shouldbe proportional to the corresponding particle bounce periodbetween mirror points in the vicinity of the bifurcation fieldline, when averaged over uniform distributions of the bounce

phases. The bounce period can be expressed as

2v

Zsm2sm1

1 B s Bm

1=2ds:

[33] The integrand approaches infinity atsm1 and sm2. By

Taylor expansion, the integral near the mirror point becomesintegrable,

2

v

Zsm1"sm1

1 1 B0

sm1 Bm

s sm1 1=2

ds 2v

Z"0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bm

B0

sm1

sdsffiffi

sp 2

v

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bm

B0

sm1

s2ffiffiffi"

p;

where B(sm1) < 0. We would get a similar result at mirrorpoint sm2. However, when this mirror point is the localmaximum near the equatorial plane, the first derivative will

Figure 16. Results of tracing the same electron as in 15 using the Northrop equations (12)(13) but withthe two extra terms, which are the same as the simulation results (Figure 4c) obtained by solving theBrizardChan equations.


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become zero. Adding the second derivative term in theTaylor expansion yields

2

vZsm2

sm2

"

1 1 B00

sm2 Bm

s sm2 22

1=2

ds 2v

Z"0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Bm

B00

sm2

sds

s;

which is an improper integral.[34] We designed an ensemble of particles similar to the

way it was done by ztrk and Wolf[2007]. Over 100 par-ticles located between mirror points along a magnetic fieldline were launched from local midnight with both upwardand downward velocities and traced through guiding centerequations. The pitch angles of these particles should satisfy

B(xi, yi, zi) = Bm sin2ai to ensure they stay on the same drift

shell. Under the several different cases we investigated, wedid not find an acceptable criterion for the ratio of particlesgoing into the Northern and Southern hemispheres; we haveto leave this question open.

7.4. DiffusionType Coefficient Associated With DriftShell Bifurcation

[35] The changes in the second adiabatic invariant eval-uated in this work clearly have implications for modeling ofradiation belts dynamics. Although it is not straightforwardto translate these calculations into more familiar and morecommonly used forms such as diffusion coefficients [e.g.,

Albert, 2009; Summers, 2005], we provide a simplemindedestimate of average changes, as follows. We would like toconsider an ensemble of electrons with different pitch angles

Figure 17. (top) Plots of (left) magnetic field magnitude and (right) the corresponding values of kalongthe full particle trajectory. The electron is launched with initial position (8, 0, 0) RE in GSM coordinates,initial velocity 0.99c, initial pitch angle 45. (bottom) Same as top, but electrons initial velocity isreduced to 0.9c. The solar wind and IMF conditions are the same as in previous numerical simulations.


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and different initial positions in space, and calculate theaverage changes hDIi, h(DI)2i, etc. (their expressions andvalues described below), over this ensemble, for the nominalcase of IMF BY = 5 nT.

[36] We place an ensemble of 418 electrons on a surfacelocated on the dawn side and obtained by interpolating frommagnetic field lines with equatorial crossing distance rang-ing from 4 to 10 RE. Then we adopt the following approach

to ensure that it is equivalent to placing a uniform density ofelectrons per unit area on the surface: we prescribe a meshof uniformly distributed points in the XGSM = 0 plane on thedawn side and assign each projected point on the surface withan electron and a weight factor proportional to the surfacearea enclosed by its four nearest neighbors; the weight factorwk (kis the electron number) is later included in the expres-sions of the ensemble averages.

[37] To set up the ensemble, we place a 341 keV electronwith a random value ofcosabetween 1and1ateachpointonthe surface. Here a is the pitch angle; cosa is uniformly dis-tributed over the range [1, 1]. The initial gradient/curvaturedrift velocity perpendicular to the surface V?gc,k is alsocalculated.

[38] The ensemble of electrons with initial positions andpitch angles prescribed above are started from the surface onthe dawn side and traced using the Brizard and Chan [1999]guiding center equations. Out of them, 66 electrons expe-rience drift shell bifurcation, being trapped either into the

Northern Hemisphere or into the Southern Hemisphere on thedayside. Their second invariant changesrIkand drift periodstkare recorded. Electrons that drift through the magnetopauseor fail to undergo drift shell bifurcation are eliminated fromthe calculation. Our results are

DIh i

Pk

wkDIkV?gc;k

PkwkV?gc;k

1:0RE;

DI 2D E

P

k

wk DIk 2V?gc;kPk

wkV?gc;k 3:2R2E;

DI

( )P

k

wk DIk=k V?gc;kPk

wkV?gc;k 0:001RE=s;

DI 2

2* + P

k

wk DIk 2=2k

V?gc;k

Pk w

kV?gc;k 0:002R2E=s;

where the last quantity can be considered a diffusiontypecoefficient. The energy dependence in the last two expres-sions should be that of the gradient/curvature drift velocity,which is proportional to gv2 (g and v are the relativisticfactor and the total velocity, respectively). So a factor of(WK / 273 keV)(1 + WK / 1022 keV) / (1 + WK / 511 keV)should be multiplied to the righthand side for differentenergies. Here WK is the electron kinetic energy. For elec-trons of the energy considered in this calculation, the ratio of

the diffusiontype coefficient to the one (0.411 rm2 (Gdawn

2 +Gdust2 ) /td; equation(49)) given by ztrk and Wolf[2007]

and evaluated for typical parameters suggested in that paperi s 2 1 04, showing that the impact due to the field asymmetryis likely to dominate.

8. Conclusions

[39] We presented analysis of the second adiabatic invari-ant changes of trapped energetic particles in the magneto-sphere due to drift shell bifurcation near the magnetopauseforthe case of magnetic fields lacking eastwest and northsouthsymmetry. For the magnetic field configurations we studied,electrons subject to the DSB effect were in the energy range ofkeV up to MeV, while protons were supposed to have lowerenergies due to their larger gyroradii and possible violationof the first adiabatic invariant conservation. We found thattypical changes in the second invariant after a particle crossesthe dayside bifurcation region are comparable to the initialvalue of the invariant, and we demonstrated the result in twoways, by solving guiding center equations to trace particletrajectories and also by analyzing the magnetic field only

(both results agree with each other). We studied dependenceof the effect on the parameters controlling the TsyganenkoT02 magnetic field model and found that, for a given value ofthe mirror field, the absolute value of the second invariantchangeDItends to depend on IMF BY and somewhat on BZand is larger with increasing values of the solar wind dynamic

pressure; the dependence on Dst is rather weak. When par-ticles come back to the same value of local time that theystarted from, if they get a net increase in the second invariant,they are at a larger radial distance and have a higher possi-

bility of getting lost to the magnetopause once they drift to thedayside. These results have implications for modeling ofradiation belts in the inner magnetosphere, although we didnot parameterize the effect in terms of loss terms for typical

radiationtype calculations. In addition, we clarified why theBrizard and Chan [1999] guiding center equations had anadvantage over the standardNorthrop [1963] equations whennumerically tracing charged particles in realistic magneto-spheric magnetic fields.

[40] Acknowledgments. This work was supported by the NASAGeospace Theory grants NNG05GH93G and NNX08AI55G.

[41] Masaki Fujimoto thanks Xin Tao and another reviewer for theirassistance in evaluating this paper.

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M. K. ztrk, Department of Information Systems and Technologies,Yeditepe University, 34755 Istanbul, Turkey.

S. Sazykin, Y. Wan, and R. A. Wolf, Physics and AstronomyDepartment, Rice University, Houston, TX, 77005 USA. ([email protected])


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drift shell bifurcation near the dayside magnetopause in realistic magnetospheric magnetic fields

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